Preconditioned HLLE Method for Flowsat All Mach Numbers

Soo Hyung Park∗

Konkuk University, Seoul 143-701, Republic of Korea

and

Jae Eun Lee† and Jang Hyuk Kwon‡

Korea Advanced Institute of Science Technology, Daejeon 305-701, Republic of Korea

DOI: 10.2514/1.12176

A two-dimensional preconditioned multigrid Navier–Stokes solver has been developed for all Mach number

viscous flows. For the computation of all Mach number flows, a local preconditioning technique with a modified

reference velocity was used to extend the Harten–Lax–van Leer–Einfeldt (HLLE�) scheme, which has good

characteristics for compressible viscousflows. The developedmethods used to compute several representative steady

problems. Grid refinement and parametric studies are performed to verify the performance. The results show that

the preconditioned HLLE� scheme can be successfully applied to a wide variety of flows and that the implicit

multigrid solver provides good convergence for the test problems.

Introduction

A number of numerical flux functions for inviscid fluxes havebeen devised as approximate solutions to Riemann problems. Hartenet al.[1] suggested a mathematical theory for the upstream differencescheme and the Godunov-type scheme, denoted by the Harten–Lax–van Leer (HLL) which approximates the solution of the Riemannproblemwith two signal waves. A typical example of the HLL solveris the HLLE scheme [2] that proposed the bounds of signal waves byusing the eigenvalues of the Roe matrix to satisfy the entropy andpositivity conditions. Despite its desirable properties, it is difficult tosimulate practical problems because of its highly dissipativebehavior. The HLLE-modified (HLLEM) scheme [3] enhances theresolution to a level comparable to that of the Roe scheme [4].However, several problems have been reported for this scheme [5,6].The HLLE� scheme [6], which successfully eliminates theerroneous dissipation and the instability of the HLLEM scheme, hasbeen devised to predict the supersonic viscous flows accurately.

Generally, the Godunov-type schemes are unsuitable for solvinglow Mach number or incompressible flow problems because of thelarge condition number of the eigensystem of the governingequations. Time-derivative preconditioning techniques have beenproposed to reduce the stiffness and convergence problems thatoccur at low Mach number flows. Since Viviand [7] proposed ageneralized preconditioning procedure for a class of hyperbolicsystems, many researchers[8–12] have conducted a number ofpreconditioning studies. By altering the acoustic wave speeds, theirmethods make the condition number bounded independently of theMach number of the flows.

Our objectives are to modify the HLLE� scheme [6], which issuitable for simulating supersonic viscous flows, and to assess theperformance of the modifiedHLLE� scheme at lowMach numbers.

Previously, Luo et al. [13] modified the signal velocities of the HLLwith contact restoration (HLLC) scheme based on the eigenvalues ofthe preconditioned system and implemented at transonic and lowspeed viscous flows. As with earlier attempts [13–15] to modifyshock-capturing schemes, we used the time-derivative precondition-ing technique of Weiss and Smith [10] and slightly modified thereference velocity to improve its robustness both at supersonic andsubsonic regions. We used the preconditioned HLLE� scheme tocompute several representative problems of low speed viscousflows.Grid refinement and parametric studies verified the results.

Numerical Methods

Time-Derivative Preconditioning

We consider the preconditioned form of the two-dimensionalcompressible Navier-Stokes equations,

�@Q

@t� @F

@x� @G

@y� @Fv

@x� @Gv

@y� g (1)

where Q is the primitive flow variable vector, F and G are theinviscid fluxes in x- and y-directions, respectively, andFv andGv arethe viscous fluxes:

Q �puvT

2664

3775; F�

�u�u2 � p�uv�uH

2664

3775; G�

�v�uv

�v2 � p�vH

2664

3775 (2)

F v � �0; �xx; �xy; u�xx � v�xy � qx�T (3)

G v � �0; �yx; �yy; u�yx � v�yy � qy�T (4)

Here �,p, andT are the density, pressure, and temperature, and u andv are the Cartesian velocity components. H � e� p=� is the totalenthalpy and e is the total energy. The quantities �ij and qi are theviscous stresses and heat fluxes in each direction. The laminarviscosity coefficient is evaluated by using the Sutherland’s law. ThePrandtl number based on reference transport properties is 0.7. gdenotes the source vector, such as a gravitational body force.Specially, pressure and temperature terms in the primitive variableset are expressed as a perturbed form to decrease roundoff errors atsubsonic flows [9,16].

The preconditioning matrix � is used to scale acoustic wavespeeds by introducing a preconditioned velocity scale. The matrix

Presented as Paper 2709 at the 34thAIAAFluidDynamics Conference andExhibit, Portland, OR, 28 June–1 July 2004; received 12 July 2004; revisionreceived 4 April 2006; accepted for publication 4 July 2006. Copyright ©2006 by the authors. Published by the American Institute of Aeronautics andAstronautics, Inc., with permission. Copies of this paper may be made forpersonal or internal use, on condition that the copier pay the $10.00 per-copyfee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923; include the code $10.00 in correspondence with the CCC.

∗Research Professor, NITRI, 1 Hwayang-dong, Gwangjin-Gu; pish@konkuk.ac.kr. Member AIAA.

†Doctoral Candidate, Department of Aerospace Engineering, 373-1Guseong-dong, Yuseong-gu; LeeJaeeun@kaist.ac.kr.

‡Professor, Department of Aerospace Engineering, 373-1 Guseong-dong,Yuseong-gu; jhkwon@kaist.ac.kr. Senior Member AIAA.

AIAA JOURNALVol. 44, No. 11, November 2006

2645

chosen is originally proposed by Choi and Merkle [9] and extendedby Weiss and Smith [10]. � is given by

�� 1=�RT� 0 0 ��=Tu��� 1=�RT�� � 0 ��u=Tv��� 1=�RT�� 0 � ��v=T

H��� 1=�RT�� � 1 �u �v ��Cp �H=T�

2664

3775 (5)

where the parameter � is defined as

�� 1

U2r

� 1

a2(6)

with a is the speed of sound.The reference velocity, Ur, has been chosen basically to avoid

unstable behaviors at near-stagnation regions and to recover theoriginal governing equations at supersonic flow regions [11,17]. Thereference velocity should not drop below the local convection ordiffusion velocity. For low Reynolds number viscous flows, thereference velocity should not be smaller than the local diffusionvelocity,�=��d, where�d is a intercell length scale. An additionallimitation onUr has been imposed to increase the numerical stability.The limitation prohibits amplification of the pressure perturbations inthe stagnation regions [12]. Therefore, the restriction onUr becomes

Ur �min

�a;max

�jUj; KU1;

�

��d; �

�����������j�pj�

s ��(7)

whereK is arbitrary but fixed at 0.5 for the global cutoff, and j�pj isobtained as the maximum for all the faces of a given cell [17]. Asunderlined in [17], this formulation with the pressure gradient sensorcan lead to unstable behavior, particularly in the presence of strongacoustic waves. The scaling parameter �, which typically rangesfrom 0.001 to 0.1, should be carefully tuned to prevent the instabilitybecause of its sensitivity, especially when a multigrid algorithm isapplied. This instability can be easily prevented by exploiting thepressure perturbation itself rather than the neighboring pressuregradients, which are expressed as follows:

Ur �min

�a;max

�jUj; KUcut;

�

��d;

��������jp0j�

s ��(8)

where Ucut �min�U1; a1� and p0 � �pl � pr�=2. pl and pr havethemeaning of the left and right gauge pressures at the adjacent cells,respectively. The optimized definition of the reference velocity,Eq. (8), can improve the robustness of an implicit multigridalgorithm without the sensitive scaling parameter �. Moreover, themodified parameter KUcut becomes small enough to activatepreconditioning in the subsonic regions dominantly supersonicflowssuch as blunt-body flows by employing freestream speed of sound.

Godunov-Type Flux-Splitting Schemes

Numerical flux functions have been constructed for one-dimensional Euler equations because the viscous fluxes can bediscretized by central differencing. Equation (1) can be integratedcellwise in one-dimensional space domain, then the discretizedgoverning equation yields in a conservative form:

W n�1i �Wn

i ��t

�x

�Fn

i�1=2 � Fni�1=2

�(9)

where W is the conservative flow variable vector,

W � � � �u �v �e �T (10)

The HLL Riemann solver [1] approximates the solution of theRiemann problemwith twowaves propagating at speeds ofbr andbl,as shown in Fig. 1. They are the lower and upper bounds for thephysical signal speeds with which the information of the initialdiscontinuity is transported. To satisfy the entropy and the positivityconditions, Einfeldt [2] suggested adequate bounds bymaking use of

theRoe-averaged eigenvalues, �p � fu; u� a; u � a; ug, where a isthe speed of sound at the cell interface and the subscriptp varies from1 to 4. The superscript^denotes the Roe-averaged values throughoutthis paper. The HLLE scheme does not violate the positivity and theentropy conditions nor suffer instability at strong shocks. However,the contact discontinuities are smeared excessively because it is verydissipative regardless of the chosen bounds. TheHLLEMscheme [3]improves the resolution of contact discontinuity by reusing theinformation of contact discontinuity.

The schemes described in the preceding section can be unified andcharacterized by using control parameters. This enables us to analyzethe dissipation mechanisms of each scheme. The numerical flux ofthe unified Godunov-type schemes is defined by

Fi�1=2 �b�F�W l� � b�F�Wr�

b� � b� � b�b�

b� � b�

��Wr �W l�

�Xp�1;4

�� �pTp

(11)

with b� �maxfbr; 0:0g and b� �minfbl; 0:0g, where br and bl aredefined to be

br �maxf�2;C�g; bl �minf�3;C

�g (12)

HereTp (p� 1, 2, 3, 4) are the right eigenvectors of theflux Jacobianevaluated at intermediate states. �p are coefficients of the projectionofWr �W l onto Tp:

W r �W l �X4p�1

�pTp (13)

Becauseb� andb� are positive and negative quantities, respectively,the last term inEq. (11) is of an antidiffusive nature. The antidiffusion

coefficient �� is defined such that they can take out excess dissipationin linear degenerated fields:

��� a

j �uj � a(14)

where the speed �u is defined as the approximate speed of the contactdiscontinuity.

According to the definition of C�, C� and �u, different dissipationmechanisms can be described as follows:

HLLE:

C� � ur � ar; C� � ul � al; j �uj �1 so that ��� 0:0

(15)

forwardnon-linear wave

backwardnon-linear wave

wl

t

xi+ 1/ 2

wr

brbl

wm

Fig. 1 The approximate solution of HLL Riemann solver.

2646 PARK, LEE, AND KWON

HLLEM:

C � � ur � ar; C� � ul � al; j �uj �br � bl

2

(16)

Roe:

C� � �2; C� � �3; j �uj � juj (17)

It is noted that the numerical dissipation mechanism can beunderstood mainly through the antidiffusion terms which areexplicitly identified. This is important in constructing a new HLLsolver which captures strong shocks without encountering shockinstability nor contaminating the inherent resolution of the scheme inviscous flows. We have proposed an accurate HLLE method whichenhances the dissipation mechanism of antidiffusion terms,especially for viscous flows. More detailed description can befound in [6]. For brevity, it is denoted by HLLE� in this paper.HLLE� scheme without any switching mechanism is described bythe following parameters:

C� � ur � ar; C� � ul � al; j �uj � juj (18)

The original version of the HLLE� scheme has a switchingmechanism based on a pressure sensor. It has been found in ournumerical experiments that the switching mechanism is no longerneeded in the framework of a properly preconditioned system.

Preconditioned HLLE� Formulation

Time-derivative preconditioning changes the eigensystem of thegoverning equations so that the eigenvalues of the system have thesame order as the local convective velocity. Shock-capturingmethods should therefore be modified with respect to thepreconditioned system. The eigenvalues of the preconditionedsystem are �0

p���1@F=@Q� � fu; u0 � a0; u0 � a0; ug, where

u0 � u�1 � ��; a0 ������������������������2u2 � U2

r

p(19)

�� 1� �Ur=a�22

(20)

TheHLLE� scheme can be modified to perform well at very lowMach numbers by replacing the signal velocities with thepreconditioned signal velocities and by replacing the eigenvectormatrices with the preconditioned matrices. The resulting HLLE�scheme can then be expressed as

Fi�1=2 �b�F�W l� � b�F�Wr�

b� � b� � b�b�

b� � b�

��Wr �W l�

�Xp�1;4

�� �p��T

p�

(21)

where

W r �W l �X4p�1

�p��T

p� (22)

and the preconditioned signal velocities br and bl:

br �maxfu0 � a0; u0r � a0

rg (23)

bl �minfu0 � a0; u0l � a0

lg (24)

��� a0

j �u0j � a0 (25)

The accuracy of the Godunov-type scheme is increased to thesecond-order by using the third-order MUSCL with the continuous

van-Albada limiter [18]. The simple central differencing is applied toobtain the variable gradients of the viscous fluxes.

It is noted that�p� and�T

p� in the Eq. (22) can be comparedwith�p

and Tp in the Eq. (13) as shown in the Appendix.

Time-Stepping Schemes

In this paper, a multigrid diagonalized alternate directionalimplicit (DADI) method [19,20] is applied to find steady-statesolutions. The following integral form of the governing equations isconsidered over a control volume V�t�:

�d

dt

ZV�t�

QdV �Z@V�t�

�F � Fv�dS � 0 (26)

An implicit scheme can be written as�I��t

@R

@Q

��Q���t��1R�qm� (27)

By using ADI method, Eq. (27) can be factorized as follows:

fI��tA��gfI��tA�g�Q���t��1R (28)

Here, A� � ��1�@R=@Q� are the Jacobian matrix of the residualvector in each direction. Finally, the preconditioned implicitformulation, (28), is expressed in a diagonalized ADI form by usingthe similarity transformation [21,22]:

T���I��tfr��0�� �r��0�

� � ��2�r�Av�g�T�1�� � T��I

��tfr��0� �r��0�

� ��2r�Av�g�T�1� ��Q���t��1R

(29)

where r mean the backward and forward difference operators and

�0i are the preconditioned eigenvalues in each direction. ��2i r�Av�

operators express the central differencing of viscous fluxes. Moredetailed description can be found in [22]. Unlike inviscid terms, thecontribution of viscous terms is not simultaneously diagonalizable,and it must be added to the implicit part only by an approximation ofspectral radius scaling:

r�Av� ���

Pr

�

�(30)

For a similar reason, the local time-stepping is applied to�t, which isdefined by the CFL number over the summation of the spectral radiiof the inviscid and viscous flux vectors:

�t� CFLPi

��ju0ij � a0� � r�Av��

(31)

A full approximate storage (FAS) multigrid algorithm [20] is usedfor the steady-state computation. Because the present V-cyclemultigrid algorithm concentrates on the acceleration of wavepropagation rather than high frequency damping, different numbersof time-stepping along with different CFL numbers are appliedaccording to the grid level where L is the grid level and L� 1 is thefinest level. In this work, two time-steps are advanced in the coarselevels. In addition, the CFL number can be increased according to thegrid level.When the CFL number of the finest level isCFL1, the CFL

number of the coarse levels is determined by CFLL � CFL1

���������L�p.

Numerical Results

To verify the preconditioned HLLE� scheme for low Machnumber flows, several well-known problems in both low and highMach number flows were computed and compared with the previousresults and experimental data. In this paper, Roe (P) andHLLE� �P�denote the preconditioned Roe and HLLE� schemes, respectively.

PARK, LEE, AND KWON 2647

Inviscid Flow Past a Bump

To examine the effect of preconditioning, we computed inviscidflows past a 10%arc bump at variousMach numbers. The coarse gridused consists of 65 points in the flow direction and 17 points in thenormal direction. For the inflow and outflow boundary conditions,applied is a simplified set of noncharacteristic conditions, whichyields almost the same results as the characteristic boundaryconditions [23].

Figure 2 shows the pressure perturbation contours of the HLLE�schemewithout preconditioning, whereas Fig. 3 shows the same datafor the HLLE� scheme with preconditioning. As shown in Fig. 2,theHLLE� scheme as well as the Roe scheme gives a contaminatedsolution at M� 0:001 and 0.1, where no visible differences arefound at M� 0:675 and 2.0. This unphysical behavior of commonshock-capturing schemes can be eliminated by implementing a lowMach number preconditioning method. The pressure distributions inFigs. 3a and 3b show nearly symmetric solutions of theHLLE� �P�scheme. The preconditioning method can effectively produce areasonable result up to M� 10�5 (not shown).

The effect of the preconditioning is obviously found in theconvergence histories shown in Figs. 4 and 5. A four-level multigridmethodwith a CFL of 3 is applied to all inviscid bump computations.The failed or very slow convergence in Fig. 4 is common in shock-capturing schemes when preconditioning is not applied. For the lowMach number cases in Fig. 5, the convergence of the HLLE� �P�scheme is nearly independent of Mach numbers. In contrast, theorder of the convergence varies with the freestream Mach number.

Figure 6 compares the pressure coefficient distribution of eachspatial scheme. The Roe (P) and the HLLE� �P� schemes producealmost the same results. The Roe and theHLLE� schemes generateunphysical solutions at M� 0:1 that are kinked at the leading andtrailing edges of the bump. Although the results for the transonic andsupersonic cases are not shown, the results from the preconditionedschemes are nearly the same as the nonpreconditioned ones.

Supersonic Blunt-Body Flow

The next test problemconcerns a 2-D supersonic inviscidflowpasta cylinder. This kind of test is used to verify the accuracy of anyspatial scheme for supersonic viscous flows [6]. A 65 65 grid isused and a freestream Mach number is 15.0. Here, the limiter at theMUSCL interpolation is not van-Albada butminmod having discretemanner of derivative in certain regions [18]. Figure 7a shows thewell-known shock instability, namely carbuncle. This instability hasforced Godunov-type schemes to use an entropy correction or aswitching mechanism [5,6]. However, the Roe (P) andHLLE� �P�

a)

b)

c)

d)

Fig. 2 HLLE� pressure perturbation contours with Mach numbera) 0.001, b) 0.1, c) 0.675, d) 2.0.

a)

b)

c)

d)

Fig. 3 HLLE� �P� pressure perturbation contours with Machnumber a) 0.001, b) 0.1, c) 0.675, d) 2.0.

Iteration

Nor

mal

ized

RM

Ser

ror

ofD

Q

200 400 600 800 100010-14

10-12

10-10

10-8

10-6

10-4

10-2

100

M = 1.0E-5M = 1.0E-3M = 0.1M = 0.675M = 2.0M = 5.0

Fig. 4 Convergence history for the inviscid bump with Mach number:

HLLE�.

2648 PARK, LEE, AND KWON

schemes do not suffer from this instability. It is noted that theHLLE� �P� scheme has no switching mechanism. In the case of theRoe (P) method, the minmod limiter contributes to the elimination ofcarbuncle rather than the preconditioning itself. It can be confirmedthat similar results are obtained by employing a part of Liou’s theory[5]without theminmod limiter. That is, the coefficient of the pressureat the numerical dissipation should be zero at the supersonic regimes.

The most important point of the present application is to confirmeffects of the proposed definition of the reference velocity Ur.�������������jp0j=�p

term in the Eq. (8) makes the preconditioning solvercalculate robustly under supersonic flow or multigrid condition. Asmentioned before,Ucut of themodified reference velocity enables thepreconditioning effect to be activated in the vicinity of locallysubsonic regions in globally supersonic flows. IfUcut usesU1 only,original global cutoff definition overcomes the local speed of soundeven at subsonic domains when the freestream Mach number is 15.However, once a1 is accompanied with U1 as the proposed globalcutoff definition, the KUcut term has smaller magnitude than otherterms including the local speed of sound. From the present numerical

experiment, KUcut [Eq. (8)] <jUj<�������������jp0j=�p

< a behind shock. Inaddition to these advantages, the modified limitations not only haveharmless solutions for other subsonic implementations, theconvergence rates of the preconditioned schemes in Fig. 8 are alsomuch better than those of the original schemes.

Consequently, preconditioning formulations are more effectivethan nonpreconditioning forms even if a Liou’s theory is a dominantsolution for the shock instability in a point view of carbuncle only.This phenomenon gives an interesting insight into the relationbetween the supersonic shock instability and the low Mach numberasymptotic. The numerical consequences, however, should beverified by an asymptotic analysis such as Guillard and Viozat’sanalysis [16].

Steady Flow in a Lid-Driven Cavity

As an example of the lowMach number viscous flows, a flow in alid-driven cavity is computed. TheMach number of themoving lid is0.001. Three Reynolds numbers of 1, 100, and 1000 are consideredand the no-slip isothermal wall boundary condition is applied with azero pressure gradient. Figure 9 displays the streamline contours forthe three Reynolds numbers. The streamline patterns show a largeprimary vortex coupled with two secondary vortices at the bottomcorners. To verify the accuracy of the solutions, the U- and V-velocities at the centerlines of the cavity are displayed in Figs. 10 and11. The results obtained from different grids agree well with thebenchmark solutions of Ghia et al. [24]. Figures 12 and 13 show theconvergence histories of the root-mean-square error of�Q, which is

Iteration

Nor

mal

ized

RM

Ser

ror

ofD

Q

200 400 600 800 100010-14

10-12

10-10

10-8

10-6

10-4

10-2

100

M = 1.0E-5M = 1.0E-3M = 0.1M = 0.675M = 2.0M = 5.0

Fig. 5 Convergence history for the inviscid bump with Mach number:

HLLE� �P�.

X/C

Cp

-0.5 0 0.5 1 1.5

-1

-0.5

0

0.5

1

Roe, M=0.1Roe(P), M=1.0E-5Roe(P), M=1.0E-3Roe(P), M=0.1HLLE+(P), M=1.0E-5HLLE+(P), M=1.0E-3HLLE+(P), M=0.1

Fig. 6 Comparison of pressure coefficient distributions on the bump

surface.

a) Roe b) HLLE+

c) Roe (P) d) HLLE+(P)

Fig. 7 Comparison of pressure contours with and without precondi-

tioning.

Iteration

No

rmal

ized

RM

Ser

ror

ofD

Q

0 5000 10000 15000 20000 2500010-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

RoeRoe(P)HLLE+HLLE+(P)

Single grid solutions

Fig. 8 Convergence history of blunt-body flow.

PARK, LEE, AND KWON 2649

normalized by the initial�Q0 when four-level multigrid with a CFLof 5 is applied. Except in a few cases, the convergence rate does notdepend on theReynolds number, the grid size, or the aspect ratio. Theconvergence rate largely depends on the definition of the referencevelocity, especially for low Mach number viscous flows. Althoughspecial treatments for viscous preconditioning have been devisedover the last decade [9], they have not yet been implemented.

Re = 1 Re = 100 Re = 1000Fig. 9 Streamline contours for flow in a lid-driven cavity.

U

Y

-0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

33 X 3349 X 4965 X 6565 X 65 (uniform)Re=100 (Ghia et al.)Re=1000 (Ghia et al.)

Fig. 10 U-velocity along vertical centerline for lid-driven cavity flow.

X

V

0 0.2 0.4 0.6 0.8 1

-0.6

-0.4

-0.2

0

0.2

0.4

33 X 3349 X 4965 X 6565 X 65 (uniform)Re=100 (Ghia et al.)Re=1000 (Ghia et al.)

Fig. 11 V-velocity along horizontal centerline for lid-driven cavity

flow.

Iteration

No

rmal

ized

RM

Ser

ror

of

DQ

200 400 600 800 1000

10-10

10-8

10-6

10-4

10-2

100

33 X 33 ( AR = 5 )49 X 49 ( AR = 3 )65 X 65 ( AR = 10 )65 X 65 (uniform)

Re = 1

Re = 100

Fig. 12 Convergence history for driven cavity forRe� 1 and 100 withgrid size.

Iteration

No

rmal

ized

RM

Ser

ror

of

DQ

200 400 600 800 1000

10-10

10-8

10-6

10-4

10-2

100

33 X 33 ( AR = 5 )49 X 49 ( AR = 3 )65 X 65 ( AR = 10 )65 X 65 (uniform)

Re = 1000

Fig. 13 Convergence history for driven cavity forRe� 1000with grid

size.

Ra = 103

TH

Ra = 105 Ra = 106

TL

Fig. 14 Streamline and isotemperature contours in thermally driven

cavity for different Rayleigh numbers.

2650 PARK, LEE, AND KWON

Steady Flow in a Thermally Driven Cavity

The next test problem is a buoyancy-drivenflow in a square cavity.The configuration comprised two vertical walls at temperature TH

andTL alongwith two adiabatic horizontal walls. The solution to thiscomplex problem depends on a variety of flow parameters, such asthe Rayleigh number [Ra� �2g��TH � TL�L3Cp=��k�], the aspectratio of the cavity, and a temperature difference parameter[�� �TH � TL�=�TH � TL�]. Here � is the thermal expansioncoefficient, g is the magnitude of the gravitational field, L is thelength of the cavity walls, and� and k are the dynamic viscosity andthermal conductivity, respectively.

For the present study, three Rayleigh numbers,Ra� 103, 105, and106, are considered with a temperature difference parameter �� 0:6.The grid sizes are 65 65, 129 129, and 257 257. Themaximum aspect ratios (AR) of each grid are 10, 40, and 120.Figure 14 displays streamlines and temperature contours for eachRayleigh number. The center of the vortex at Ra� 103 shiftedtoward the lower and cold wall. A recirculating roll is driven by thegeneration of vorticity because of the horizontal temperaturegradient. At Ra� 105 and 106 the secondary vortices are generatedbecause of the adverse temperature gradient [9].

For each Rayleigh number case, a grid refinement study isperformed in line with the approach of Vierendeels et al. [15]. Theresults are summarized in Table 1. The extrapolated values arecomputed with Richardson’s extrapolation method:

fextrap � fh � Ch� (32)

�� ln �fh � fh=2�=�fh=2 � fh=4�ln �2� (33)

C� fh � fh=2h��1 � 2��� (34)

where h is given by 1=N and,N is the number of grid cells in a singledirection. We used h� 1=257. The computed � values show thequadratic grid convergence. As shown in Fig. 15, the extrapolatedNusselt values are in an excellent agreement with the correlationgiven by Chenoweth and Paolucci [25].

The convergence histories for each grid size are displayed inFig. 16. Although the convergence rate does not depend largely onthe Rayleigh number for each grid, the rate slows as the grid aspectratio increases. The implicit multigrid method mainly causes thisdependence but a special viscous preconditioning [9] can alleviatethe convergence degradation.

Conclusions

We extended the HLLE� scheme in conjunction with time-derivative preconditioning for low Mach number viscous flows.Limitations on the reference velocity must be carefully imposed ontoa preconditioned scheme, especially when it is developed forflows atall Mach numbers. Proposed additional limitations using pressureperturbation andmodified global cutoff terms are shown to lead to animprovement of the robustness of the preconditioning system. Theinviscid bump and cylinder problems were tested to assess theperformance of our scheme. Parametric studies for viscous cavityproblems reveal that the present multigrid algorithm gives a goodconvergence characteristic that is essentially independent ofReynolds numbers and Rayleigh numbers.

Appendix: Preconditioned Matrices

Following formulations [10,16,22] include components of theantidiffusive term for the original and preconditioned HLLE�schemes.

HLLE� method

An inviscid Jacobian A and a conserved variable set W can bedefined as follows when the governing equation is two-dimensionalEuler equations.

Table 1 Nusselt numbers on the hot wall extrapolated with

Richardson’s method with grid size

N Nu, Ra� 103 Nu, Ra� 105 Nu, Ra� 106

65 1.0974143 4.4675517 8.7412005129 1.0936196 4.4278758 8.5951033257 1.0906887 4.4140846 8.5586672extrap. 1.0807433 4.4067367 8.5465609� �0:3726170 �1:5245111 �2:0034899C 0.0012579 1:5566613E � 06 1:7977670E � 07

Rayleigh number

Nus

selt

num

ber

102 103 104 105 106 107 1080

4

8

12

16

20

PresentCorrelation, Chenoweth & Paolucci

Fig. 15 Comparison of the extrapolated Nusselt number with a

correlation.

Iteration

Nor

mal

ized

RM

Ser

ror

ofD

Q

500 1000 1500 2000 250010-13

10-11

10-9

10-7

10-5

10-3

10-1

Ra = 10E3Ra = 10E5Ra = 10E6

Grid 65 X 65

Grid 257 X 257

Grid 129 X 129

Fig. 16 Convergence history for thermally driven cavity with grid size.

PARK, LEE, AND KWON 2651

W � � �u �v �e� �

T A�0 �x �y 0

�x� � 1� � uU U � � � 2��xu �yu � � � 1��xv � � 1��x�y� � 1� � vU �xv � � � 1��yu U � � � 2��yv � � 1��y� � 1� U �HU H�x � � � 1�uU H�y � � � 1�vU U

2664

3775

� �u2 � v2�=2; U � u�x � v�y; �� �;

A JacobianmatrixA is related with the eigensystem constituted withthe right eigenvectors and eigenvalues.

A � T�T�1

where

��A� �D�U;U � a��2x � �2y�;U � a��2x � �2y�;U�

�

U 0 0 0

0 U � a��2x � �2y� 0 0

0 0 U � a��2x � �2y� 0

0 0 0 U

266664

377775

Each element of the right eigenvectors T�p� (p� 1, 2, 3, 4) and ��p�

the coefficients of the projection of �W��Wr �W l� onto T�p� inEq. (11) can be written as follows as well:

��1���W� ��� ��p

a2; ��2���W� � 1

2a

��p

a� �� ~U

�

��3���W� � 1

2a

��p

a� �� ~U

�; ��4���W� � �� ~V

T�1��W� �1

uv

2664

3775; T�2��W� �

1

u� a ~�xv� a ~�yH � a ~U

2664

3775

T�3��W� �1

u � a ~�xv � a ~�yH � a ~U

2664

3775; T�4��W� �

0

� ~�y~�x~V

2664

3775

Here,

~U � u ~�x � v ~�y; ~V ��u ~�y � v ~�x; ~�x � �x=�����������������x � �y

p

Preconditioned HLLE� Method

In the case of the preconditioned HLLE� scheme, thepreconditioning matrix � directly affects the eigenstructure. Whenthe preconditioned Jacobian matrix A� is defined to be

A � � ��1A��U �T��x �T��y 0�x=� U 0 0�y=� 0 U 0

U��1����1��

� � 1�T��x � � 1�T��y U

2664

3775

�� U2r=a

2

then the following relation is satisfied

A � � T���T�1�

and eigenvalues and eigenvectors can be written as follows:

� ��A�� �D�U; u0 � a0; u0 � a0;U� �D

�U;

�1 � ��U2

���������������������������������������������������������������1 � ��2U2=2� U2

r

��2x � �2y

�r;U

�

T� �

0~U�� ~�

�3��

2 ~a0~��2��

� ~U�

2 ~a0 0

0 ~�x2� ~a0 � ~�x

2� ~a0 � ~�y

0~�y

2� ~a0 � ~�y2� ~a0 ~�x

1��1� ~U�� ~�

�3��

2� ~a0��1� ~��2�

�� ~U�

2� ~a0 0

2666664

3777775

~��2�� � ~u0 � ~a0; ~�

�3�� � ~u0 � ~a0

��p�� and T�p�

� can be also induced such as the original HLLE�scheme.

��1�� ��W� � �T

�

��� ��p

U2r

�; ��2�

� ��W� � r

��p

r� �� ~U

�

��3�� ��W� � s

��p

s� �� ~U

�; ��4���W� �� ~V

�T�1�� �W� � � �

T

1

uv

2664

3775; �T�2�

� �W� � 1

r2 ~a0

1

u� r ~�xv� r ~�yH � r ~U

2664

3775

�T�3�� �W� � 1

s2 ~a0

1

u � s ~�xv � s ~�yH � s ~U

2664

3775; �T�4�

� �W� � �

0

� ~�y~�x~V

2664

3775

where

r� U2r

~U� � ~��3��

; s� U2r

~��2�� � ~U�

References

[1] Hartern, A., Lax, P. D., and Van Leer, B., “On Upstream Differencingand Godunov-Type Schemes for Hyperbolic Conservation Laws,”SIAM Review, Vol. 25, No. 1, 1983, pp. 35–61.

[2] Einfeldt, B., “On Godunov-Type Methods for Gas Dynamics,” SIAMJournal on Numerical Analysis, Vol. 25, No. 2, 1988, pp. 294–318.

[3] Einfeldt, B., Munz, C. D., Roe, P. L., and Sjögreen, B., “On Godunov-Type Methods Near Low Densities,” Journal of Computational

Physics, Vol. 92, No. 2, 1991, pp. 273–295.[4] Roe, P. L., “Characteristic Based Schemes for the Euler Equation,”

Annual Review of Fluid Mechanics, Vol. 18, 1986, pp. 337–365.[5] Liou, M.-S., “Mass Flux Schemes and Connection to Shock

Instability,” Journal of Computational Physics, Vol. 160, No. 2, 2000,pp. 623–648.

[6] Park, S. H., and Kwon, J. H., “On the Dissipation Mechanism ofGodunov-Type Schemes,” Journal of Computational Physics,Vol. 188, No. 2, 2003, pp. 524–542.

[7] Viviand, H., “Pseudo-Unsteady Systems for Steady Inviscid FlowCalculations,” Numerical Methods for the Euler Equations of Fluid

2652 PARK, LEE, AND KWON

Dynamics, edited by F. Angrand, A. Dervieux, J. A. Desideri, and R.Glowinski, SIAM, Philadelphia, 1985, p. 334.

[8] Van Leer, B., Lee,W. T., andRoe. P. L., “Characteristic Time-Steppingor Local Preconditioning of the Euler Equations,” AIAA Paper 91-1552, June 1991.

[9] Choi, Y.-H., andMerkle, C. L., “The Application of Preconditioning inViscous Flows,” Journal of Computational Physics, Vol. 105, No. 2,1993, pp. 207–223.

[10] Weiss, J. M., and Smith, W. A., “Preconditioning Applied to Variableand Constant Density Flows,” AIAA Journal, Vol. 33, No. 11, 1995,pp. 2050–2057.

[11] Turkel, E., “Preconditioning Techniques in Computational FluidDynamics,” Annual Review of Fluid Mechanics, Vol. 31, 1999,pp. 385–416.

[12] Weiss, J. M., Maruszewski, J. P., and Smith, W. A., “Implicit Solutionof Preconditioned Navier-Stokes Equations Using Algebraic Multi-grid,” AIAA Journal, Vol. 37, No. 1, 1999, pp. 29–36.

[13] Luo, H., Baum, J. D., and Lohner, R., “Extension of Harten-Lax-vanLeer Scheme for Flows at All Speeds,” AIAA Journal, Vol. 43, No. 6,2005, pp. 1160–1166.

[14] Edwards, J. R., and Liou, M.-S., “Low-Diffusion Flux-SplittingMethods for Flows at All Speeds,” AIAA Journal, Vol. 36, No. 9, 1998,pp. 1610–1617.

[15] Vierendeels, J.,Merci, B., andDick, E., “BlendedAUSM�Method forAll Speeds andAll GridAspect Ratios,”AIAA Journal, Vol. 39, No. 12,2001, pp. 2278–2282.

[16] Guillard, H., and Viozat, C., “On the Behaviour of Upwind Schemes inthe Low Mach Number Limit,” Computers & Fluids, Vol. 28, No. 1,1999, pp. 63–86.

[17] Venkateswaran, S., Li, D., and Merkle, C. L., “Influence of Stagnation

Regions on Preconditioned Solutions at Low Speeds,” AIAAPaper 2003-0435, Jan. 2003.

[18] Anderson,W.K., Tomas, J. L., andVanLeer, B., “Comparison of FiniteVolume Flux Vector Splittings for the Euler Equations,”AIAA Journal,Vol. 24, No. 9, 1986, pp. 1453–1460.

[19] Pulliam, T. H., and Chaussee, D. S., “A Diagonal Form of an ImplicitApproximate-Factorization Algorithm,” Journal of Computational

Physics, Vol. 39, No. 2, 1981, pp. 347–363.[20] Ni, R.-H., “AMultiple-Grid Scheme for Solving the Euler equations,”

AIAA Journal, Vol. 20, No. 11, 1982, pp. 1565–1571.[21] Dailey, L. D., and Pletcher, R. H., “Evaluation of Multigrid

Acceleration for Preconditioned Time-Accurate Navier-StokesAlgorithms,” Computers & Fluids, Vol. 25, No. 8, 1996, pp. 791–811.

[22] Jespersen, D., Pulliam, T., and Buning, P., “Recent Enhancements toOVERFLOW,” AIAA Paper 97-0644, Jan. 1997.

[23] Turkel, E., Radespiel, R., and Kroll, N., “Assessment ofPreconditioning Methods for Multidimensional Aerodynamics,”Computers & Fluids, Vol. 26, No. 6, 1997, pp. 613–634.

[24] Ghia, U., Ghia, K. N., and Shin, C. T., “High-Re Solutions forIncompressible Flow Using the Navier-Stokes Equations and aMultigrid Method,” Journal of Computational Physics, Vol. 48, No. 3,1982, pp. 387–411.

[25] Chenoweth, D. R., and Paolucci, S., “Natural Convection in anEnclosed Vertical Air Layer with Large Horizontal TemperatureDifferences,” Journal of Fluid Mechanics, Vol. 169, Aug. 1986,pp. 173–210.

G. CandlerAssociate Editor

PARK, LEE, AND KWON 2653